Optimal. Leaf size=148 \[ \frac {32 b^3 \tanh ^{-1}(\tanh (a+b x))^{7/2}}{3003 x^{7/2} \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^4}+\frac {16 b^2 \tanh ^{-1}(\tanh (a+b x))^{7/2}}{429 x^{9/2} \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^3}+\frac {2 \tanh ^{-1}(\tanh (a+b x))^{7/2}}{13 x^{13/2} \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )}+\frac {12 b \tanh ^{-1}(\tanh (a+b x))^{7/2}}{143 x^{11/2} \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^2} \]
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Rubi [A] time = 0.08, antiderivative size = 148, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {2171, 2167} \[ \frac {16 b^2 \tanh ^{-1}(\tanh (a+b x))^{7/2}}{429 x^{9/2} \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^3}+\frac {32 b^3 \tanh ^{-1}(\tanh (a+b x))^{7/2}}{3003 x^{7/2} \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^4}+\frac {2 \tanh ^{-1}(\tanh (a+b x))^{7/2}}{13 x^{13/2} \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )}+\frac {12 b \tanh ^{-1}(\tanh (a+b x))^{7/2}}{143 x^{11/2} \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^2} \]
Antiderivative was successfully verified.
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Rule 2167
Rule 2171
Rubi steps
\begin {align*} \int \frac {\tanh ^{-1}(\tanh (a+b x))^{5/2}}{x^{15/2}} \, dx &=\frac {2 \tanh ^{-1}(\tanh (a+b x))^{7/2}}{13 x^{13/2} \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )}+\frac {(6 b) \int \frac {\tanh ^{-1}(\tanh (a+b x))^{5/2}}{x^{13/2}} \, dx}{13 \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )}\\ &=\frac {12 b \tanh ^{-1}(\tanh (a+b x))^{7/2}}{143 x^{11/2} \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^2}+\frac {2 \tanh ^{-1}(\tanh (a+b x))^{7/2}}{13 x^{13/2} \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )}+\frac {\left (24 b^2\right ) \int \frac {\tanh ^{-1}(\tanh (a+b x))^{5/2}}{x^{11/2}} \, dx}{143 \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^2}\\ &=\frac {16 b^2 \tanh ^{-1}(\tanh (a+b x))^{7/2}}{429 x^{9/2} \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^3}+\frac {12 b \tanh ^{-1}(\tanh (a+b x))^{7/2}}{143 x^{11/2} \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^2}+\frac {2 \tanh ^{-1}(\tanh (a+b x))^{7/2}}{13 x^{13/2} \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )}+\frac {\left (16 b^3\right ) \int \frac {\tanh ^{-1}(\tanh (a+b x))^{5/2}}{x^{9/2}} \, dx}{429 \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^3}\\ &=\frac {32 b^3 \tanh ^{-1}(\tanh (a+b x))^{7/2}}{3003 x^{7/2} \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^4}+\frac {16 b^2 \tanh ^{-1}(\tanh (a+b x))^{7/2}}{429 x^{9/2} \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^3}+\frac {12 b \tanh ^{-1}(\tanh (a+b x))^{7/2}}{143 x^{11/2} \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^2}+\frac {2 \tanh ^{-1}(\tanh (a+b x))^{7/2}}{13 x^{13/2} \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 82, normalized size = 0.55 \[ \frac {2 \tanh ^{-1}(\tanh (a+b x))^{7/2} \left (-1001 b^2 x^2 \tanh ^{-1}(\tanh (a+b x))+819 b x \tanh ^{-1}(\tanh (a+b x))^2-231 \tanh ^{-1}(\tanh (a+b x))^3+429 b^3 x^3\right )}{3003 x^{13/2} \left (\tanh ^{-1}(\tanh (a+b x))-b x\right )^4} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.49, size = 78, normalized size = 0.53 \[ \frac {2 \, {\left (16 \, b^{6} x^{6} - 8 \, a b^{5} x^{5} + 6 \, a^{2} b^{4} x^{4} - 5 \, a^{3} b^{3} x^{3} - 371 \, a^{4} b^{2} x^{2} - 567 \, a^{5} b x - 231 \, a^{6}\right )} \sqrt {b x + a}}{3003 \, a^{4} x^{\frac {13}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.20, size = 97, normalized size = 0.66 \[ -\frac {\sqrt {2} {\left (\frac {429 \, \sqrt {2} b^{13}}{a} - 2 \, {\left (\frac {143 \, \sqrt {2} b^{13}}{a^{2}} + 4 \, {\left (\frac {2 \, \sqrt {2} {\left (b x + a\right )} b^{13}}{a^{4}} - \frac {13 \, \sqrt {2} b^{13}}{a^{3}}\right )} {\left (b x + a\right )}\right )} {\left (b x + a\right )}\right )} {\left (b x + a\right )}^{\frac {7}{2}} b}{3003 \, {\left ({\left (b x + a\right )} b - a b\right )}^{\frac {13}{2}} {\left | b \right |}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.39, size = 151, normalized size = 1.02 \[ -\frac {2 \arctanh \left (\tanh \left (b x +a \right )\right )^{\frac {7}{2}}}{13 \left (\arctanh \left (\tanh \left (b x +a \right )\right )-b x \right ) x^{\frac {13}{2}}}-\frac {12 b \left (-\frac {\arctanh \left (\tanh \left (b x +a \right )\right )^{\frac {7}{2}}}{11 \left (\arctanh \left (\tanh \left (b x +a \right )\right )-b x \right ) x^{\frac {11}{2}}}-\frac {4 b \left (-\frac {\arctanh \left (\tanh \left (b x +a \right )\right )^{\frac {7}{2}}}{9 \left (\arctanh \left (\tanh \left (b x +a \right )\right )-b x \right ) x^{\frac {9}{2}}}+\frac {2 b \arctanh \left (\tanh \left (b x +a \right )\right )^{\frac {7}{2}}}{63 \left (\arctanh \left (\tanh \left (b x +a \right )\right )-b x \right )^{2} x^{\frac {7}{2}}}\right )}{11 \left (\arctanh \left (\tanh \left (b x +a \right )\right )-b x \right )}\right )}{13 \left (\arctanh \left (\tanh \left (b x +a \right )\right )-b x \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.42, size = 56, normalized size = 0.38 \[ \frac {2 \, {\left (16 \, b^{4} x^{4} - 40 \, a b^{3} x^{3} + 70 \, a^{2} b^{2} x^{2} - 105 \, a^{3} b x - 231 \, a^{4}\right )} {\left (b x + a\right )}^{\frac {5}{2}}}{3003 \, a^{4} x^{\frac {13}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.74, size = 413, normalized size = 2.79 \[ \frac {\sqrt {\frac {\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )}{2}-\frac {\ln \left (\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )}{2}}\,\left (\frac {27\,b\,x\,\left (\ln \left (\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )-\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )+2\,b\,x\right )}{143}-\frac {106\,b^2\,x^2}{429}-\frac {{\left (\ln \left (\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )-\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )+2\,b\,x\right )}^2}{26}+\frac {20\,b^3\,x^3}{3003\,\left (\ln \left (\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )-\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )+2\,b\,x\right )}+\frac {16\,b^4\,x^4}{1001\,{\left (\ln \left (\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )-\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )+2\,b\,x\right )}^2}+\frac {128\,b^5\,x^5}{3003\,{\left (\ln \left (\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )-\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )+2\,b\,x\right )}^3}+\frac {512\,b^6\,x^6}{3003\,{\left (\ln \left (\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )-\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )+2\,b\,x\right )}^4}\right )}{x^{13/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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